Chemins antipodaux discrets et continus de surface incurvée. Extensions du théorème de Borsuk-Ulam et application de l'intégrale du chemin de Feynman et de l'intégrale du contour de Wodehouse
Bibliographic record
Abstract
This paper introduces discrete and continuous paths over simply-connected surfaces with non-zero curvature as means of comparing and measuring paths between antipodes with either a Feynman path integral or Wodehouse contour integral, resulting in a number of extensions of the Borsuk Ulam Theorem. Methods: All paths originate on a Riemannian surface S, which is simplyconnected and has non-zero curvature at every point in S. A surface is a simply connected, provided every cross-cut divides the surface into disjoint regions. A cross cut is an arc that runs through the interior of the surface S without self-intersections and joins one boundary to another. Results: A fundamental result in this paper is that for any pair of antipodal surface points, a path that begins and ends at the antipodal points can be found. This result is extended to Feynman path integrals on trajectory-ofparticle curves and to N.M.J. Wodehouse countour integrals for antipodal vectors on twistor curves. Another fundamental result in this paper is that the Feynman trajectory of a particle is realizable as a Lefschetz arc.
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How this classification was reachedexpand
Full frame distilled prediction
Teacher imitationNot calibrated prevalence, not ground truth. Human validation pending. Learned from the 10,348 direct Codex labels and 10,348 direct Gemma labels. Candidate is the union of thresholded teacher heads; consensus is their intersection. These outputs are machine_predicted_unvalidated and are not human labels or direct frontier model labels.
Codex and Gemma teacher scores by category
| Category | Codex | Gemma |
|---|---|---|
| Metaresearch | 0.024 | 0.009 |
| Meta-epidemiology (narrow) | 0.001 | 0.001 |
| Meta-epidemiology (broad) | 0.001 | 0.000 |
| Bibliometrics | 0.000 | 0.000 |
| Science and technology studies | 0.001 | 0.000 |
| Scholarly communication | 0.000 | 0.000 |
| Open science | 0.002 | 0.002 |
| Research integrity | 0.001 | 0.002 |
| Insufficient payload (model declined to judge) | 0.000 | 0.000 |
Machine scores (provisional)
The two teacher heads of the student model, read on this work. A score orders the frame for review; it never asserts a category, and the validation status ships verbatim with every row.
Baseline scores from an immature model (maturity gate not passed, 7 training rounds). Scores rank; they never assert a category.
score_only:v0-immature-baseline · verbatim from the scoring run: score_only means the number may rank works, and no category label ships from itClassification
machine, unvalidatedMachine predicted; a candidate call from one teacher head, not a consensus.
How this classification was reached, model by model and score by score, is at the end of the page under "How this classification was reached".